1.10 Recursive Definition

To prove \(P(x)\) holds for all Q#1 in recursively defined set \(R\), prove:

• \(P(\)Q#2\()\) for each base case \(b\) in Q#3

• \(P(\)Q#4\()\) for each constructor, \(c\), assuming induction hypothesis \(P(\)Q#5\()\)

1. Which variable should replace the placeholder Q#1?

   Exercise 1

     \(b\) \(b(x)\) \(c\) \(c(x)\) \(R\) \(x\) \(x\) 

2. Which variable should replace the placeholder Q#2?

   Exercise 2

     \(b\) \(b(x)\) \(c\) \(c(x)\) \(R\) \(x\) \(b\) 

3. Which variable should replace the placeholder Q#3?

   Exercise 3

     \(b\) \(b(x)\) \(c\) \(c(x)\) \(R\) \(x\) \(R\) 

4. Which variable should replace the placeholder Q#4?

   Exercise 4

     \(b\) \(b(x)\) \(c\) \(c(x)\) \(R\) \(x\) \(c(x)\) 

5. Which variable should replace the placeholder Q#5?

   Exercise 5

     \(b\) \(b(x)\) \(c\) \(c(x)\) \(R\) \(x\) \(x\) 
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